Equivalent latent model will be, \[ \begin{bmatrix} \mathbf{w} \\ \mathbf{z} \end{bmatrix} \sim \mathbf{N}\left( \begin{bmatrix} \boldsymbol{\mu}_{w} \\ \boldsymbol{\mu}_x \end{bmatrix}, \begin{bmatrix} \boldsymbol{\Sigma}_{ww} & \boldsymbol{\Sigma}_{wz} \\ \boldsymbol{\Sigma}_{zw} & \boldsymbol{\Sigma}_{zz} \end{bmatrix} \right) \]
How much should I discuss about simrel-M??
n): 100m): 3relpos): 1, 4; 2, 3ypos): 1; 2, 3p): 2 levels (20, 250)gamma): 2 levels (0.2, 0.9)eta): 2 levels (0.1, 0.8)R2: 2 levels (0.8, 0.8; 0.4, 0.8)Methods used in the study and their short description (how they estimate, what are they based on)
As Xenv, Yenv and Senv are based on maximum likelihood estimation, principal components of predictors explaining 99.5% of their variation are used.
In the model the prediction error for each of three response variables are used as response variable and following variables (with levels) and their complete interactions are used as predictor variables.
Following is the MANOVA output for estimation error and prediction error models using number of components (tuning parameters) that results minimum error.
Estimation Error Model:
Analysis of Variance Table
Df Pillai approx F num Df den Df Pr(>F)
(Intercept) 1 0.85222 12052.7 3 6270 < 2.2e-16 ***
p 1 0.09529 220.1 3 6270 < 2.2e-16 ***
gamma 1 0.68418 4527.6 3 6270 < 2.2e-16 ***
eta 1 0.38443 1305.2 3 6270 < 2.2e-16 ***
R2 1 0.05217 115.0 3 6270 < 2.2e-16 ***
Method 7 0.88602 375.5 21 18816 < 2.2e-16 ***
p:gamma 1 0.00396 8.3 3 6270 1.648e-05 ***
p:eta 1 0.00613 12.9 3 6270 2.187e-08 ***
gamma:eta 1 0.16480 412.4 3 6270 < 2.2e-16 ***
p:R2 1 0.00577 12.1 3 6270 6.556e-08 ***
gamma:R2 1 0.04252 92.8 3 6270 < 2.2e-16 ***
eta:R2 1 0.00009 0.2 3 6270 0.9011113
p:Method 7 0.21428 68.9 21 18816 < 2.2e-16 ***
gamma:Method 7 0.78221 316.0 21 18816 < 2.2e-16 ***
eta:Method 7 0.40078 138.2 21 18816 < 2.2e-16 ***
R2:Method 7 0.12490 38.9 21 18816 < 2.2e-16 ***
p:gamma:eta 1 0.00060 1.3 3 6270 0.2862542
p:gamma:R2 1 0.00040 0.8 3 6270 0.4739464
p:eta:R2 1 0.00031 0.7 3 6270 0.5806291
gamma:eta:R2 1 0.00017 0.4 3 6270 0.7874492
p:gamma:Method 7 0.07353 22.5 21 18816 < 2.2e-16 ***
p:eta:Method 7 0.02282 6.9 21 18816 < 2.2e-16 ***
gamma:eta:Method 7 0.23322 75.5 21 18816 < 2.2e-16 ***
p:R2:Method 7 0.01715 5.2 21 18816 1.125e-13 ***
gamma:R2:Method 7 0.09159 28.2 21 18816 < 2.2e-16 ***
eta:R2:Method 7 0.00190 0.6 21 18816 0.9409893
p:gamma:eta:R2 1 0.00003 0.1 3 6270 0.9830458
p:gamma:eta:Method 7 0.00613 1.8 21 18816 0.0112070 *
p:gamma:R2:Method 7 0.00793 2.4 21 18816 0.0003875 ***
p:eta:R2:Method 7 0.00359 1.1 21 18816 0.3696117
gamma:eta:R2:Method 7 0.00307 0.9 21 18816 0.5684497
p:gamma:eta:R2:Method 7 0.00308 0.9 21 18816 0.5651083
Residuals 6272
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Prediction Error Model:
Analysis of Variance Table
Df Pillai approx F num Df den Df Pr(>F)
(Intercept) 1 0.99958 5027601 3 6270 < 2.2e-16 ***
p 1 0.06360 142 3 6270 < 2.2e-16 ***
gamma 1 0.43493 1609 3 6270 < 2.2e-16 ***
eta 1 0.99826 1201956 3 6270 < 2.2e-16 ***
R2 1 0.99118 234856 3 6270 < 2.2e-16 ***
Method 7 0.87097 367 21 18816 < 2.2e-16 ***
p:gamma 1 0.03254 70 3 6270 < 2.2e-16 ***
p:eta 1 0.00083 2 3 6270 0.155656
gamma:eta 1 0.07579 171 3 6270 < 2.2e-16 ***
p:R2 1 0.00353 7 3 6270 6.063e-05 ***
gamma:R2 1 0.04235 92 3 6270 < 2.2e-16 ***
eta:R2 1 0.00021 0 3 6270 0.726919
p:Method 7 0.20968 67 21 18816 < 2.2e-16 ***
gamma:Method 7 0.41067 142 21 18816 < 2.2e-16 ***
eta:Method 7 0.14667 46 21 18816 < 2.2e-16 ***
R2:Method 7 0.26131 85 21 18816 < 2.2e-16 ***
p:gamma:eta 1 0.00200 4 3 6270 0.005629 **
p:gamma:R2 1 0.00669 14 3 6270 3.855e-09 ***
p:eta:R2 1 0.00027 1 3 6270 0.641880
gamma:eta:R2 1 0.00188 4 3 6270 0.008216 **
p:gamma:Method 7 0.20889 67 21 18816 < 2.2e-16 ***
p:eta:Method 7 0.01058 3 21 18816 1.262e-06 ***
gamma:eta:Method 7 0.03617 11 21 18816 < 2.2e-16 ***
p:R2:Method 7 0.07252 22 21 18816 < 2.2e-16 ***
gamma:R2:Method 7 0.09446 29 21 18816 < 2.2e-16 ***
eta:R2:Method 7 0.00529 2 21 18816 0.043606 *
p:gamma:eta:R2 1 0.00129 3 3 6270 0.044494 *
p:gamma:eta:Method 7 0.01106 3 21 18816 4.137e-07 ***
p:gamma:R2:Method 7 0.04514 14 21 18816 < 2.2e-16 ***
p:eta:R2:Method 7 0.00442 1 21 18816 0.146295
gamma:eta:R2:Method 7 0.00578 2 21 18816 0.020300 *
p:gamma:eta:R2:Method 7 0.00321 1 21 18816 0.509954
Residuals 6272
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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Study of Effect